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Jun 18, 20261
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Researchers Develop Skew Polynomial Framework for Division Algebras and Maximum Rank Distance Codes
A new arXiv submission presents a skew polynomial framework for constructing division algebras and linear maximum rank distance matrix codes. The framework generalizes several prominent mathematical constructions including those of Sheekey, Jha-Johnson, and Albert, with applications to coding theory and information theory.
Quick Facts
Who
Researchers (names not specified in submission)
What
Constructed division algebras using skew polynomials
When
Submitted 16 June 2026
Where
arXiv (arxiv.org)
- Constructed division algebras using skew polynomials
- Developed linear maximum rank distance (MRD) matrix codes
- Generalized Sheekey's twisted cyclic pre-semifields
- Generalized Jha-Johnson semifields
- Generalized Albert's generalized twisted fields
Researchers have submitted a new mathematical framework to arXiv that constructs division algebras and linear maximum rank distance (MRD) matrix codes using skew polynomials over fields. The work, categorized under Computer Science and Information Theory, presents a comprehensive approach to generating non-unital division algebras that extends several important existing constructions in the field.
The framework generalizes multiple prominent mathematical structures, including Sheekey's twisted cyclic pre-semifields, the pre-semifields associated with Jha-Johnson semifields, and the semifields derived from Albert's generalized twisted fields. Additionally, the linear MRD codes presented in the research generalize earlier constructions developed by Lobillo, Santonastaso, and Sheekey, demonstrating the broader applicability of the new approach.
The authors provide explicit criteria for determining when the constructed algebras qualify as division algebras and when the codes achieve maximum rank. The research includes a comparative analysis of isotopic division algebras found throughout recent and classical mathematical literature, along with computations of their key invariants. This work contributes to the theoretical understanding of algebraic structures relevant to coding theory and information theory applications.
Topics
Why This Matters
This research advances fundamental mathematical structures with direct applications to coding theory and information security. The unified framework enables researchers to construct and analyze division algebras and maximum rank distance codes more systematically, potentially improving error-correcting codes used in communications and data storage systems. The generalization of multiple classical constructions provides a deeper theoretical foundation for understanding algebraic structures in coding and information theory.
Timeline & Sources
Jun 16, 2026
WireResearch paper submitted to arXiv
Jun 18, 2026
WirePaper published on arXiv with identifier cs.IT (Computer Science > Information Theory)